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(Math) Sorting: Pancake Sorting

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Joined: 06 Jul 2005
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Location: Angel C. de Dios

PostPosted: Mon Sep 04, 2006 7:14 am    Post subject: (Math) Sorting: Pancake Sorting Reply with quote

Science News Online
Week of Sept. 2, 2006; Vol. 170, No. 10

Pancake Sorting
Ivars Peterson

The cook at the Sunrise Cafe is sloppy. When she prepares pancakes, they all come out a different size. The server, however, is tidy. Before he delivers a stack to a customer, he rearranges the pancakes in order of size, with the smallest one on top and the largest on the bottom. To do so, he grabs several pancakes from the top and flips them over. He repeats this "grab-and-flip" operation as many times as necessary, varying the number of pancakes that he flips each time. If he has a stack of n pancakes, what's the maximum number of flips that he'll ever need to use to rearrange them?

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Hiding Information in the Order of Things

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Joined: 06 Jul 2005
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Location: Angel C. de Dios

PostPosted: Wed Jul 18, 2007 11:00 am    Post subject: Fuzzy world Reply with quote

Fuzzy world
STAR SCIENCE By Raymond R. Tan, Ph.D.
Thursday, July 19, 2007

“As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics.” – Lotfi Zadeh, 1973

In 1965, Lotfi Zadeh published his seminal paper on fuzzy set theory. It took years of gestation before the rest of the world started to grasp the practical implications of this mathematical theory. Today, articles on fuzzy set theory and applications are routinely published in various scientific journals in fields that include engineering, decision theory, artificial intelligence and mathematics. There is even one journal — Fuzzy Sets and Systems — which has focused exclusively on this topic since its initial publication in 1978. At the same time, a large number of modern everyday devices make use of intelligent control based on fuzzy sets. For example, the air conditioner in my car has a fuzzy controller; sometimes it seems to me that it is too “smart” for my own convenience.

What exactly is a fuzzy set? Basically, it is a set for which partial membership is possible. In contrast, conventional sets (“crisp” sets, in fuzzy terminology) allow only for complete membership or non-membership. Consider, for example, the contrasting cases of the sets of “people in their 20s” (a crisp set, since a person is either in his or her 20s, or not) and “young people” (a fuzzy set, whose borders are not so distinct). This simple distinction gives rise to a wide array of possibilities. Rules of inference can be formulated using fuzzy logic to develop control algorithms for highly non-linear, multiple input/output systems, as for the smart air conditioner I mentioned previously. Such devices may be difficult to regulate using classical control theory. Similarly, fuzzy rules can be used as a basis for expert systems — computer programs with specialized, domain-specific knowledge comparable to human experts and capable of functioning as “virtual consultants.” Other applications include fuzzy optimization and possibility theory (as opposed to the more well-known probability theory).

Some of my research deals with the use of optimization models to design clean, efficient processes or industrial ecosystems. This work typically involves formulating mathematical models to optimize certain quantities (e.g., minimize the use of water resources) subject to constraints formulated as equations or inequalities that describe the properties of the system. These mathematical models are simplified representations of reality. They can then be manipulated using computers in order to give insights on how real systems should be built in order to reduce environmental impacts of industrial activities. It stands to reason that the validity of these insights hinges on how faithfully the models capture reality.

It is often the case that the model-building process requires a crisp mathematical formulation of real systems for which precise data is unavailable. Crisp mathematical constraints, for example a restriction of energy usage in an industrial process to 20 megawatts or less, are often counterintuitive. In this case, 20 megawatts is perfectly acceptable, while 20.1 megawatts is completely unacceptable. This sudden transition from acceptable to unacceptable is a core feature of many mathematical models — just like the proverbial straw that breaks the camel’s back. There is a certain intuitive appeal in allowing for a gradual transition from an acceptable to unacceptable solutions — and this is possible using fuzzy optimization models. Such models yield more robust and realistic solutions.

In a related problem, it is sometimes necessary to compare different technologies in order to select the environmentally preferable option — and in many cases such comparisons must be done with incomplete or imprecise data. For example, it is can be quite difficult to predict the exact fuel economy of cars two or three decades from now. Such imprecise data can be characterized using a branch of fuzzy sets called possibility theory. Fuzzy numbers with “possibility distributions” can be used to model imprecise quantities, especially when probability theory is inappropriate. Furthermore, it is possible to quantify “degrees of preference” when comparing two or more different quantities.

Space restrictions prevent me from discussing theoretical aspects and practical implications of fuzzy mathematics in this column. However, it is possible to draw some useful lessons on the nature of scientific research from the history of fuzzy sets. It provides a useful counterpoint to the common notion research must be fully utilitarian, or that the research community must work exclusively on urgent issues of practical nature. This way of thinking may work in the short term, but in the end it will be counterproductive for science and technology. After all, Zadeh did not set out to develop a mathematical theory for control systems, optimization or decision theory in his 1965 paper. These applications came much later. What he did was develop the theoretical foundations that eventually led to all sorts of practical applications in the decades that followed, and which continues to find new and surprising uses to the present day.

* * *

Dr. Raymond R. Tan is an associate professor of the Chemical Engineering Department and researcher exemplar at De La Salle University-Manila. A recipient of multiple awards from the National Academy of Science and Technology, some of his work on the use of fuzzy computing has been published in the International Journal of Energy Research, Computers and Chemical Engineering, and Environmental Modelling and Software. For more information, visit his website at
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PostPosted: Sat Aug 11, 2007 7:23 am    Post subject: Cracking the Cube Reply with quote

Week of Aug. 11, 2007; Vol. 172, No. 6

Cracking the Cube
Solving Rubik's Cube gets one move quicker
Julie J. Rehmeyer

Daniel Kunkle can solve a Rubik's Cube in 26 moves. Or at least his computer can.

Kunkle, a computer scientist at Northeastern University in Boston, has proved that 26 moves are enough to solve any Rubik's Cube, no matter how scrambled. That's one move below the previous record. In the process of cracking the cube, he developed algorithms that can be useful for problems as disparate as scheduling air flights and determining how proteins will fold.

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